🚀 go-pugleaf

I take John Collins's point about the problem with skew distributions
being with the data, not with the correlation coefficients - but only
some of the way ....  Of course, it isn't only with Pearson correlations
betwen dichotomous items (which surely *are* Phi coefficients, aren't
they??) that skewness can distort the outcome of a factor analysis,
since the theory assumes Gaussian distributions.  Focussing on the situation
where the items are dichotomous, and are skewed, it would be very nice
if any emergence of "difficulty factors" were guaranteed to make its
presence known in the form of loadings > 1   (Heywood cases). However,
on the basis of trying some simulations, I don't think this can be
guaranteed to occur.  I've tried variants on the following procedure
to help me grasp the issue of "spurious" factors emerging from
factor analyses:

1.  Generate a normal distribution of simulated scores on some putative
    latent trait.
2.  From this, generate two sets of simulated observable variables, Set A
    and Set B, such that scores on each of the variables in Set A is
    jointly determined by
    i) the answer to the question "Is your latent trait Z-score
       greater than -0.3, and
    ii) a bit of randomness, thrown in to stop them all correlating with
        each other.

    Variables in Set B are generated similarly, except that the question is
    "Is your latent trait Z-score greater that +0.3 ?"

There is no doubt that there is only one latent trait which underlies all
the variables, but variables within the two sets are similar in their
degrees of skewness.  Every time I have tried factor analysing such
simulated data, the result has been at least two factors with eigenvalues
greater than 1, and at least two factors by the Scree Test - and I have
never had factor loadings appear with values greater than 1.

Tony Downing                               A.C.Downing @ newcastle.ac.uk
Dept. of Psychology,
University of Newcastle upon Tyne.